A rigorous definition of axial lines: ridges on isovist fields

Rui Carvalho (1) and Michael Batty(2)

rui.carvalho@ucl.ac.uk m.batty@ucl.ac.uk

(1) The Bartlett School of Graduate Studies

(2) Centre for Advanced Spatial Analysis

University College London,

1-19 Torrington Place, London WC1E 6BT, UK

Abstract We suggest that ‘axial lines’ defined by (Hillier and Hanson, 1984) as lines of

uninterrupted movement within urban streetscapes or buildings, appear as ridges

in isovist fields (Benedikt, 1979). These are formed from the maximum diametric

lengths of the individual isovists, sometimes called viewsheds, that make up these

fields (Batty and Rana, 2004). We present an image processing technique for the

identification of lines from ridges, discuss current strengths and weaknesses of the

method, and show how it can be implemented easily and effectively.

Introduction: from local to global in urban morphology Axial lines are used in space syntax to simplify connections between spaces that

make up an urban or architectural morphology. Usually they are defined

manually by partitioning the space into the smallest number of largest convex

subdivisions and defining these lines as those that link these spaces together.

Subsequent analysis of the resulting set of lines (which is called an ‘axial map’)

enables the relative nearness or accessibility of these lines to be computed. These

can then form the basis for ranking the relative importance of the underlying

spatial subdivisions and associating this with measures of urban intensity, density,

or traffic flow. To date, progress has been slow at generating these lines

automatically. Lack of agreement on their definition and lack of awareness as to

how similar problems have been treated in fields such as pattern recognition,

robotics and computer vision have inhibited explorations of the problem and only

very recently have there been any attempts to evolve methods for the automated

generation of such lines (Batty and Rana, 2004; Ratti, 2001). One obvious advantage of a rigorous algorithmic definition of axial lines is the

potential use of the computer to free humans from the tedious tracing of lines on

large urban systems. Perhaps less obvious is the insight that mathematical

procedures may bring about urban networks, and their context in the burgeoning

body of research into the structure and function of complex networks (Albert and

Barabási, 2002; Newman, 2003). Indeed, on one hand urban morphologies display

a surprising degree of universality (Batty and Longley, 1994; Carvalho and Penn,

2003; Frankhauser, 1994; Makse et al., 1995; Makse et al., 1998) but little is yet

known about the transport and social networks embedded within them (but see

(Chowell et al., 2004)). On the other hand, axial maps are a substrate for human

navigation and rigorous extraction of axial lines may substantiate the

development of models for processes that take place on urban networks which

range from issues covering the efficiency of navigation, the way epidemics

propagate in cities, and the vulnerability of network nodes and links to failure,

attack and related crises. Further, axial maps are discrete models of continuous

systems and one would like to understand the consequences of the transition to a

discrete approach.

In what follows, we hypothesise a method for an algorithmic definition of axial

lines inspired by local properties of space, which eliminates both the need for us

to define convex spaces and to trace “(…) all lines that can be linked to other

axial lines without repetition” (Hillier and Hanson, 1984, p 99). A definition of

axial lines (global entities) with neighbourhood methods (local entities) implies

that transition from small to large-scale urban environments carries no new

theoretical assumptions and that the computational effort grows linearly (less

optimizations) with the number of mesh points used. Our main goal is to gain

insight into urban networks in general and axial lines in particular. Therefore we

leave algorithm optimizations for future work. It is, however, beyond the scope of

the present note to address generalizations of axial maps or to integrate current

theories with GIS (but see (Batty and Rana, 2004; Jiang et al., 2000)).

The method: Axial lines as ridges on isovist fields Axial maps can be regarded as members of a larger family of axial

representations (often called skeletons) of 2D images. There is a vast literature on

this, originating with the work of Blum on the Medial Axis Transform (MAT)

(Blum, 1973; Blum and Nagel, 1978), which operates on the object rather than

its boundary (see (Tonder et al., 2002) for a link between Visual Science and the

MAT applied to a Japanese Zen Garden). Geometrically, the MAT uses a

circular primitive. Objects are described by the collection of maximal discs, ones

which fit inside the object but in no other disc inside the object. The object is the

logical union of all of its maximal discs. The description is in two parts: the locus

of centres, called the symmetric axis and the radius at each point, called the

radius function, R (Blum and Nagel, 1978). The MAT employs an analogy to a

grassfire. Imagine an object whose border is set on fire. The subsequent internal

convergence points of the fire represent the symmetric axis, the time of

convergence for unity velocity propagation being the radius function (Blum and

Nagel, 1978).

An isovist is the space defined around a point (or centroid) from which an object

can move in any direction before the object encounters some obstacle. In space

syntax, this space is often regarded as a viewshed and a measure of how far one

can move or see is the maximum line of sight through the point at which the

isovist is defined. We shall see that the paradigm shift from the set of maximal

discs inside the object (as in the MAT) to the maximal straight line that can be

fit inside its isovists holds a key to understanding what axial lines are.

As in space syntax, we simplify the problem by eliminating terrain elevation and

associate each isovist centroid with a pair of horizontal coordinates ( ),x y and a

third coordinate - the length of the longest straight line across the isovist at each

point which we define on the lattice as max,i j∆ where ( ),x y is uniquely associated

with ),( ji . Our hypothesis states that all axial lines are ridges on the surface of max,i j∆ . The reader can absorb the concept by “embodying” herself in the max

,i j∆

landscape: movement along the perpendicular direction to an axial line implies a

decrease along the max,i j∆ surface; and max

,i j∆ is an invariant, both along the axial

line and along the ridge. Our hypothesis goes further to predict that the converse

is also true, i.e., that up to an issue of scale, all ridges on the max,i j∆ landscape are

axial lines. Most of what follows is the development of a method to extract these

ridges from the max,i j∆ surface, in the same spirit that one would process

temperature values sampled spatially with an array of thermometers.

Our method follows a procedure similar to the Medial Axis Transform (MAT).

Indeed, the MAT approach to skeletonization first calculates a scalar field for the

object (the Distance Map) and then identifies a set of ridge points, or generalized

local maxima, in this scalar map. In a discretized representation, the final

skeleton consists of such ridge points with the possible addition of a set of points

necessary to form a connected structure (Simmons and Séquin, 1998).

Here we sample isovist fields by generating isovists for the set of points on a

regular lattice (Batty, 2001; Ratti, 2001; Turner et al., 2001). This procedure is

standard practice in spatial modelling (Burrough and McDonnell, 1998).

Specifically, we are interested in the isovist field defined by the length of the

longest straight line across the isovist at each mesh point, ( ),i j . This measure is

denoted the maximum diametric length, max,i j∆ (Batty and Rana, 2004), or the

maximum of the sum of the length of the lines of sight in two opposite directions

(Ratti, 2001, p 204). To simplify notation, we will prefer the former term.

First, we generate a Digital Elevation Model (DEM) (Burrough and McDonnell,

1998) of the isovist field, where max,i j∆ is associated with mesh point ( ),i j (Batty,

2001; Ratti, 2001). Next, we use a point algorithm to locate the ridges based on

their convexity that is orthogonal to a line with no convexity/concavity (Rana and

Morley, 2002) on the DEM. Our algorithm detects ridges by extracting the local

maxima of the discrete DEM. Next, we use an image processing transformation

(the Hough Transform) on a binary image containing the local maxima points

which lets us rank the detected lines in the Hough parameter space. Finally, we

invert the Hough transform to find the location of axial lines on the original

image.

The Hough transform (HT) was developed in connection with the study of

particle tracks through the viewing field of a bubble chamber (the detection

scheme was first published as a patent of an electronic apparatus for detecting

the tracks of high-energy particles). It was one of the first attempts to automate

a visual inspection task previously requiring hundreds of man-hours to execute

(Leavers, 1993) and is used in computer vision and pattern recognition for

detecting geometric shapes that can be defined by parametric equations. Related

applications of the HT include detection of road lane markers (Kamat-Sadekar

and Ganesan, 1998; Pomerleau and Jochem, 1996) and determination of urban

texture directionality (Habib and Kelley, 2001; Ratti, 2001).

The HT converts a difficult global detection problem in image space into a more

easily solved local peak detection problem in parameter space (Illingworth and

Kittler, 1988). The basic concept involved in locating lines is point-line duality.

In an influential paper, Duda and Hart (Duda and Hart, 1972) suggested that

straight lines might be usefully parameterized by the length, ρ , and orientation,

θ , of the normal vector to the line from the image origin. Imagine that there is a

ridge line in image space. The normal vector for each point on this line is defined

by cos sinx yρ θ θ= + where ρ and θ are the same for any pair of coordinates

( ),x y . If we then compute all lines passing through each pair of coordinates on

the ridge line in terms of their normal vector, count all the length and orientation

parameters ( ),ρ θ , and then plot these counts in the parameter space defined by

ρ and θ , the position of each straight line (ridge) in image space will be marked

as a peak in parameter space. This then enables us to define the locations of

ridges in image space simply from examining all possible normal vectors for all

possible points. In short, each point ( ),P x y= in the image space is mapped into

a sinusoidal curve in the ( ),ρ θ space, cos sinx yρ θ θ= + , and points lying on

the same straight line in the image plane correspond to curves through a common

point in the parameter plane—see Figure 1. The HT specifies a line as follows.

Imagine yourself standing on the image plane at the origin of the coordinates,

facing the positive y direction —see Figure 1c). Turn a specified angle, Lθ , to your

right, and then walk a specified number of pixels forward, Lρ . Turn through 90°

and go forward; you are now walking along the required line in the image.

The process of using the HT to detect lines in an image involves the computation

of the HT for the entire image, accumulating evidence in an array for events by a

voting (counting) scheme (points in the parameter plane “vote” for the

parameters of the lines to which they possibly belong) and searching the

accumulator array for peaks which hold information of potential lines present in

the input image. The peaks provide only the length of the normal to the line and

the angle that the normal makes with the y -axis. They do not provide any

information regarding the length, position or end points of the line segment in

the image plane (Gonzalez and Woods, 1992). Our line detection algorithm starts

by extracting the point that has the largest number of votes on parameter space,

which corresponds to the line defined by the largest number of collinear local

maxima of max,i j∆ , and proceeds by extracting lines in rank order of the number of

their votes on parameter space. One of us has previously proposed (Batty and

Rana, 2004) rank-order methods as a rigorous formulation of the procedure

originally outlined of “first finding the longest straight line that can be drawn,

then the second longest line and so on (…)” (Hillier and Hanson, 1984, p 99).

0 25 50 75 1000

25

50

75

100

X

YPoint P

P

ρθ

θ

ρ

Hough Transform of P

45 90 135 180

−50

−25

0

25

50

0 25 50 75 1000

25

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ρθ

X

Y

θ

ρ

Hough Transform of line through P

(ρ L,θ

L)

45 90 135 180

−50

−25

0

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50

L

L

O

O

y

x

x

y P

Line segment through P

a) b)

c) d)

Figure 1. a) Point ( )75,75P = in the image plane. b) The HT converts P into a

sinusoidal curve, cos sinx yρ θ θ= + , where ( ) ( ), 25, 25x y = are the coordinates

of P relative to ( )50,50O = and [ ]0,180θ ∈ . c) Line segment between points

( )0, 50 and ( )50, 0 . This segment crosses point P and is orthogonal to the

segment OP . d) The line segment in c) is identified in Hough space by the point

where all the sinusoids intersect, ( ) ( ) ( ), 2 , 45 35.4, 45L L xρ θ = ≅ . The line defined

by the segment in c) can be rebuilt on the image plane by starting at O facing

the direction of the positive y axis, turning Lθ degrees to the right, walking

forward Lρ (until P ) and finally tracing the perpendicular line to OP .

To test the hypothesis that axial lines are equivalent to ridges on the max,i j∆

surface, we start with a simple geometric example: an ‘H’ shaped open space

structure (see Figure 2). As illustrated in Figure 2, axial lines are equivalent to

ridges for this simple geometric example, if extended until the borders on the

open space.

Figure 2. (a) Plot of the Maximum Diametric Length ( max

,i j∆ ) isovist field for an

‘H’ shaped open space structure. (b) Zoom-in (detail) of (a) showing the ridges

on the longer arms of the ‘H’ shape. Arrows point to the ridges on both figures.

Indeed, one confirms this both in Figure 2a) and Figure 2b) by properly zooming-

in the max,i j∆ landscape. Next, we aim at developing a method to extract these

ridges as lines by sampling. In Figure 3a), we plot the local maxima of the

discretized max,i j∆ landscape, which are a discretized signature of the ridges on the

max,i j∆ continuous field. Figure 3b) is the Hough transform of Figure 3a) where θ

goes from 0° to 180° in increments of 1°. The peaks on Figure 3b) are the

maxima in parameter space, ( ),ρ θ , which are ranked by height in Figure 3c). The

first four visible peaks in parameter space —Figures 3b) and 3c)— correspond to

the four symmetric lines defined by the highest number of collinear points in the

original space —Figure 3a).

Figure 3. (a) Local maxima of the Maximum Diametric Length ( max

,i j∆ ) for the ‘H’

shaped structure in Fig. 1. (b) Hough transform of (a). (c) Rank of the local

maxima of the surface in (b). (d) The Hough transform is inverted and the 6

highest peaks in (c) define the axial lines shown.

Finally, the ranked maxima in parameter space are inverted onto the coordinates

of the lines in the original space, yielding the detected lines which are plotted on

Figure 3d) where we only plot the lines corresponding to the 6 highest peaks in

parameter space. Having tested the hypothesis on a simple geometry, we repeat the procedure for

the French town of Gassin —see Figure 4. We have scanned the open space

structure of Gassin (Hillier and Hanson, 1984, p 91) as a binary image and

reduced the resolution of the scanned image to 300 dpi (see inset of Figure 4).

0

20

40

60

80

100 0

10

20

30

40

50

60

0200400

Figure 4. Plot of the Maximum Diametric Length ( max

,i j∆ ) isovist field for the

town of Gassin. The inset shows the scanned image from “The Social Logic of

Space” (Hillier and Hanson, 1984).

The resulting image has 171×300 points, and is read into a Matlab matrix. Next

we use a ray-tracing algorithm in Matlab (angle step=0.01°) to determine the max,i j∆ measure for each point in the mesh that corresponds to open space. The

landscape of max,i j∆ is plot on Figure 4. The next step is to extract the ridges on

this landscape. To do this, as we have seen before, we determine the local

maxima on the max,i j∆ landscape. Next, we apply the Hough Transform as in the

‘H’ shape example and invert it to determine the 6 first axial lines for the town of

Gassin (see Figure 5). We should alert readers to the fact that as we have not

imposed any boundary conditions on our definition of lines from the Hough

Transform, three of these lines intersect building forms illustrating that what the

technique is doing is identifying the dominant linear features in image space but

ignoring any obstacles which interfere with the continuity of these linear features.

We consider that this is a detail that can be addressed in subsequent

development of the approach.

Figure 5. (a) Axial lines for the town of Gassin (Hillier and Hanson, 1984). (b)

Local maxima of max,i j∆ (squares) and lines detected by the proposed algorithm.

Discussion: where do we go from here? Most axial representations of images aim at a simplified representation of the

original image, in graph form and without the loss of morphological information.

Therefore, most Axial Shape Graphs are invertible –a characteristic not shared

with Axial Maps, as the original shape cannot be uniquely reconstructed from the

latter. Also, metric information on the nodes length is often stored together with

the nodes (the latter often being weighted by the former), whereas it is

discharged in Axial Maps. On the other hand, most skeletonizations aim at a

representation of shape as the human observer sees it and therefore aim mostly at

small scale shapes (images), whereas the process of generating axial maps

assumes that the observer is immersed in the shape and aims at the

representation of large scale shapes (environments). Nevertheless, we have shown

that the extraction of axial lines can be accomplished with methods very similar

to those routinely employed in pattern recognition and computer vision (e.g. the

Medial Axial Transform and the Hough Transform). Our hypothesis has successfully passed the test of extracting axial lines both for a

simple geometry and for a traditional case study in Space Syntax – the town of

Gassin. Indeed, 2, 3, 4, 5, 6,, , , and detected detected detected detected detectedl l l l l in Figure 5 all match

reasonably well lines originally drawn (Hillier and Hanson, 1984). Differences

between original and detected lines appear for 3,originall and 3,detectedl , where the mesh

we used to detect lines was not fine enough to account for the detail of the

geometry and the HT counts collinear points along a line that intersects buildings,

and for 5,originall and 5,detectedl , where the original solution is clearly not the longest

line through the space.

Figure 5 highlights two fundamental issues which are shared by any spatial

problem, both related to the issue of tracing “all lines that can be linked to other

axial lines without repetition” (Hillier and Hanson, 1984, p 99). The first is that

defining axial lines as the longest lines of sight may lead to unconnected lines on

the urban periphery. The problem is quite evident with line 1,originall in Figure 5a)

(Hillier and Hanson, 1984, p 91), where the solution to the longest line crossing

the space is 1,detectedl —see Figure 5b). This is an expected feature of any spatial

problem, in the same way that the existence of solutions to differential equations

depends on the given boundary conditions. A possible solution may seem to be to

extend the border until lines intersect; nevertheless this may lead both to more

intersections than envisioned and disproportionate boundary sizes, as all non-

parallel lines will intersect on finite points, but not necessarily near the

settlement. Thus, the price to pay for a rigorous algorithm may be that not all

expected connections are traced. The second problem is an issue of scale, as one

could continue identifying more local ridges with increasing image resolution (see

discussion in (Batty and Rana, 2004)). We believe that the problem is solved if

the width of the narrowest street is selected as a threshold for the length of axial

lines detected from ridges on isovist fields. Only lines with length higher than the

threshold are extracted. We speculate that this satisfies almost always the

condition that all possible links are established, but are aware that more lines

will be extracted automatically than by human-processing (although it does seem

that global graph measures will remain largely unaffected by this). Again, this

seems to be the price to pay for a rigorous algorithm.

By being purely local, our method gives a solution to the global problem of

tracing axial maps in a time proportional to the number of mesh points. This

means that algorithm optimization is akin to local optimization (mesh placement

and ray-tracing algorithm). Although most of the present effort has been in

testing the hypothesis, it is obvious that regular grids are largely redundant.

Indeed, much optimization could be accomplished by generating denser grids near

points where the derivative of the boundary is away from zero (i.e., curves or

turns) to improve detection at the extremities of axial lines. Also, the algorithm

could be improved by generating iterative solutions that would increase grid and

angle sweep resolutions until a satisfactory solution would be reached or by

parallelizing visibility analysis calculations (Mills et al., 1992).

Our approach to axial map extraction is preliminary as the HT detects only line

parameters while axial lines are line segments. Nevertheless, there has been

considerable research effort put into line segment detection in urban systems,

generated mainly by the detection of road lane markers (Kamat-Sadekar and

Ganesan, 1998; Pomerleau and Jochem, 1996), and we are confident that further

improvements involve only existing theory.

This note shows that global entities in urban morphology can be defined with a

purely local approach. We have shown that there is no need to invoke the

concept of convex space to define axial lines. By providing rigorous algorithms

inspired by work in pattern recognition and computer vision, we have started to

uncover problems implicit in the original definition (disconnected lines at

boundary, scale issues), but have proposed working solutions to all of them which,

we believe will enrich the field of space syntax and engage other disciplines in the

effort of gaining insight into urban morphology. Finally, we look with

considerable optimism to the automatic extraction of axial lines and axial maps

in the near future and believe that for the first time in the history of space

syntax, automatic processing of medium to large scale cities may be only a few

years away from being implemented on desktop computers.

Acknowledgments

RC acknowledges generous financial support from Grant EPSRC GR/N21376/01

and is grateful to Profs Bill Hillier and Alan Penn for valuable comments. The

authors are indebted to Sanjay Rana for using his Isovist Analyst Extension (see

http://www.casa.ucl.ac.uk/sanjay/software_isovistanalyst.htm) to provide

independent corroboration on the ‘H’ test problem.

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